Given that sin x = 1 / sqrt(3) where x is an acute angle, determine the following: i) cos x, ii) cos2(90-x)…

Question

Given that sin x = 1 / sqrt(3) where x is an acute angle, determine the following: i) cos x, ii) cos2(90-x), iii) cot 2x. Starting from a right-angled triangle, determine whether sin x is a... Evaluate the following without use of figures.

ScanSolve AI Tutor · Accepted Answer

Dennis, hebu tusuluhishe hili.

Tunapewa $\sin x = \frac{1}{\sqrt{5}}$ na $x$ ni pembe kali.

a) (i) Tafuta $\cos x$

Tumia utambulisho wa Pythagorean $\sin^2 x + \cos^2 x = 1$ .
Badilisha thamani ya $\sin x$ :
$\left(\frac{1}{\sqrt{5}}\right)^2 + \cos^2 x = 1$ $\frac{1}{5} + \cos^2 x = 1$
Tatua kwa $\cos^2 x$ .
$\cos^2 x = 1 - \frac{1}{5}$ $\cos^2 x = \frac{4}{5}$
Tafuta $\cos x$ .
Kwa kuwa $x$ ni pembe kali, $\cos x$ itakuwa chanya.
$\cos x = \sqrt{\frac{4}{5}}$ $\cos x = \frac{2}{\sqrt{5}}$
Ili kurahisisha, zidisha nambari na denominata kwa $\sqrt{5}$ :
$\cos x = \frac{2\sqrt{5}}{5}$
Jibu la (i) ni $\boxed{\frac{2\sqrt{5}}{5}}$ .

a) (ii) Tafuta $\cos^2 (90^\circ - x)$
Tumia utambulisho wa kikoa $\cos(90^\circ - x) = \sin x$ .
Hivyo, $\cos^2 (90^\circ - x) = (\sin x)^2$ .
Badilisha thamani ya $\sin x$ .
$\cos^2 (90^\circ - x) = \left(\frac{1}{\sqrt{5}}\right)^2$ $\cos^2 (90^\circ - x) = \frac{1}{5}$
Jibu la (ii) ni $\boxed{\frac{1}{5}}$ .

a) (iii) Tafuta $\cot 2x$
Kwanza, tafuta $\tan x$ .
Tunajua $\tan x = \frac{\sin x}{\cos x}$ .
$\tan x = \frac{\frac{1}{\sqrt{5}}}{\frac{2}{\sqrt{5}}}$ $\tan x = \frac{1}{2}$
Tumia fomula ya $\tan 2x$
$\tan 2x = \frac{2 \tan x}{1 - \tan^2 x}$ .
Badilisha thamani ya $\tan x$ :
$\tan 2x = \frac{2 \left(\frac{1}{2}\right)}{1 - \left(\frac{1}{2}\right)^2}$ $\tan 2x = \frac{1}{1 - \frac{1}{4}}$ $\tan 2x = \frac{1}{\frac{3}{4}}$ $\tan 2x = \frac{4}{3}$
Tafuta $\cot 2x$ .
Tunajua $\cot 2x = \frac{1}{\tan 2x}$ .
$\cot 2x = \frac{1}{\frac{4}{3}}$ $\cot 2x = \frac{3}{4}$
Jibu la (iii) ni $\boxed{\frac{3}{4}}$ .

Given that sin x = 1 / sqrt(3) where x is an acute angle, determine the following: i) cos x, ii) cos2(90-x), iii) cot 2x. Starting from a right-angled triangle, determine whether sin x is a... Evaluate the following without use of figures.

Step-by-step solution

Tumia utambulisho wa Pythagorean $\sin^2 x + \cos^2 x = 1$ .

Tatua kwa $\cos^2 x$ .

Tafuta $\cos x$ .

Tumia utambulisho wa kikoa $\cos(90^\circ - x) = \sin x$ .

Badilisha thamani ya $\sin x$ .

Kwanza, tafuta $\tan x$ .

Tumia fomula ya $\tan 2x$

Tafuta $\cot 2x$ .

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More Mathematics Questions

Given that sin x = 1 / sqrt(3) where x is an acute angle, determine the following: i) cos x, ii) cos2(90-x), iii) cot 2x. Starting from a right-angled triangle, determine whether sin x is a... Evaluate the following without use of figures.

Step-by-step solution

Tumia utambulisho wa Pythagorean sin⁡2x+cos⁡2x=1\sin^2 x + \cos^2 x = 1sin2x+cos2x=1.

Tatua kwa cos⁡2x\cos^2 xcos2x.

Tafuta cos⁡x\cos xcosx.

Tumia utambulisho wa kikoa cos⁡(90∘−x)=sin⁡x\cos(90^\circ - x) = \sin xcos(90∘−x)=sinx.

Badilisha thamani ya sin⁡x\sin xsinx.

Kwanza, tafuta tan⁡x\tan xtanx.

Tumia fomula ya tan⁡2x\tan 2xtan2x

Tafuta cot⁡2x\cot 2xcot2x.

Still stuck on this one?

More Mathematics Questions

Tumia utambulisho wa Pythagorean $\sin^2 x + \cos^2 x = 1$ .

Tatua kwa $\cos^2 x$ .

Tafuta $\cos x$ .

Tumia utambulisho wa kikoa $\cos(90^\circ - x) = \sin x$ .

Badilisha thamani ya $\sin x$ .

Kwanza, tafuta $\tan x$ .

Tumia fomula ya $\tan 2x$

Tafuta $\cot 2x$ .